[FOM] A new geometry (in Klein's sense)?

[Dana Scott]

Giovanni Lagnese asked on Thu Feb 2 12:29:30 EST 2006:

> I would ask if the geometry (in Klein's sense) of bijective
> applications f such that (considering as example a metrical
> space (M,d) and four arbitrary elements a,b,c,d of M)
>
>     d(a,b)<d(c,d) if and only if d(f(a),f(b))<d(f(c),f(d))
>
> has been ever studied?  I think that such a geometry is too
> important to be possible that it has been never studied.

I think it is easy to prove that when M = R^n and the distance
is the usual n-dimensional Euclidean metric, then the group
of such mappings is just the usual group rigid motions of
Euclidean space.  (Maybe this works for the standard hyperbolic
and elliptic spaces as well.)  But did Giovanni have in mind
other metric spaces?

Note that functions that preserve the inequality are also
distance preserving.  In Euclidean space, distance preserving
transformations also preserve betweenness, and so preserve
the inequality.